Section 5.1 1a) zeros are x = 3 multiplicity even x = -1 multiplicity odd 1b) touches the x-axis at x = 3, crosses the x-axis at x = -1 1c) has at most 2 turning points 1d) Will behave like g(x) = x3 1e) falls to the left, rises to the right 1f) sketch a graph and approximate the turning points, also label the x-intercepts 1g) increasing (−∞, .33) ∪ (3, ∞) decreasing (.33, 3) 3a) zeros are x = 3/2 has even multiplicity and x = 1 odd multiplicity 3b) graph touches x-axis at x = 3/2 and crosses x-axis at x = 1 3c) has at most 2 turning points 3d) g(x) = 4x3 3e) falls to the left, rises to the right 3f) window used x-min -3 x-max 3 ymin -1 ymax 1 Section 5.1 continued3g) increasing (−∞, 1.17) ∪ (1.5, ∞) decreasing (1.17, 1.5) Section 5.1 continued 5a) 𝑥 = √3 ℎ𝑎𝑠 𝑒𝑣𝑒𝑛 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦, 𝑥 = −√3 ℎ𝑎𝑠 𝑒𝑣𝑒𝑛 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦, 𝑥 = 5b) graph touches x-axis at √3 𝑎𝑛𝑑 − √3 𝑐𝑟𝑜𝑠𝑠𝑒𝑠 𝑥 − 𝑎𝑥𝑖𝑠 𝑎𝑡 𝑥 = −1 2 ℎ𝑎𝑠 𝑜𝑑𝑑 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦 −1 2 5c) will have at most 4 turning points 5d) will behave like g(x) = 2x5 5e) Falls to the left rises to the right Window x-min -10 x-max 10 y-min – 10 y-max 20 5f) 5g) increasing: (−∞, −√3) ∪ (−1, .6) ∪ (√3, ∞) decreasing (−√3, −1) ∪ (.6, √3) 7a) 𝑥 = √6 ℎ𝑎𝑠 𝑒𝑣𝑒𝑛 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦, 𝑥 = −√6 ℎ𝑎𝑠 𝑒𝑣𝑒𝑛 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦, 𝑥 = 6 ℎ𝑎𝑠 𝑒𝑣𝑒𝑛 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑖𝑐𝑖𝑡𝑦 7b) Graph touches x-axis at all x-intercepts 7c) has at most 5 turning points 7d) will behave like g(x) = 4x6 7e) rises to the left and rises to the right Window used X min -10 X max 10 Y min -1000 Y max 10000 7f) 7g) : 𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 (−√6, −.45) ∪ (√6, 4.45) ∪ (6, ∞)𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔 (−∞, −√6) ∪ (−.45, √6) ∪ (4.45,6) Section 5.1 continued 9a) 9b) 9c) 9d) 9e) x = -1/2, even multiplicity graph touches x-axis at x = -1/2 has at most 3 turning points will behave like g(x) = 4x4 rises to the left and rises to the right 9f) 9g) 1 1 answer: increasing (− 2 , ∞) decreasing (−∞, − 2) Section 5.1 continued 11a) x = 6, even multiplicity 11b) Graph touches the x-axis at x = 6 11c) has at most 3 turning points 11d) will behave like g(x) = 4x4 11e) rises to the left, rises to the right 11f) g) increasing (6, ∞) decreasing (−∞, 6) 13a) 13b) 13c) 13d) 13e) 13f) x = -7, odd multiplicity x = 1, odd multiplicity graph crosses x-axis at x =-7 and at x = 1 Answer has at most 1 turning point. like g(x) = x2 rise to left and rise to the right 13g) increasing (−3, ∞) decreasing (−∞, −3) Section 5.1 continued 15a) x = -2 odd multiplicity x = 2 odd multiplicity 15b) graph crosses x-axis at both x = 2 and x = -2 15c) will have at most 1 turning point. 15d) will behave like g(x) = x2 15e) rises to the left and rises to the right 15f) 15g) increasing (0, ∞) decreasing (−∞, 0)17a) x = 0 multiplicity) 17b) graph will cross x-axis at each 0. 17c) has at most 2 turning points. 17d) will behave like g(x) = -5x3 for large values of x. 17e) rise to the left fall to the right x = -3/5 x = -2 (all zeros have odd Section 5.1 continued 17f) 17g) decreasing (−∞, −1.45) ∪ (−.27, ∞) increasing (-1.45, -.27) 19a) 19b) 19c) 19d) 19e) 19f) x = 0 has even multiplicity, x = -2 and x = 2 have odd graph touches at 0 and crosses at -2 and crosses at 2 has at most 3 turning points graph will behave like g(x) = -3x4 falls to the left, falls to the right 19g) increasing (−∞, −1.41) ∪ (0,1.41) decreasing (-1.41, 0) (1.41, ∞) Section 5.1 continued 21a)x = 0 has even multiplicity, x = -3/5 has odd and x = - 2 has odd 21b) graph touches x-axis at x = 0 and crosses x-axis at x = -3/5 and crosses at x = -2 211c) it has at most 3 turning points 21d) graph will behave like g(x) = 5x4 21e) rise to left and rise to the right 21f) sketch a graph and approximate the turning points, also label the x-intercepts 21g) increasing (-1.57, - .38) ∪ (0, ∞) decreasing (−∞, −1.57) ∪ (−.38,0) 23) f(x) = x3 – x2 – 14x + 24 27) f(x) = 3x2 – 8x – 3 25) f(x) = 2x3 -11x2 + 17x – 6 29) f(x) = x2 – 2 31) f(x) = x3 – 5x2 – 2x + 10 Section 5.2 1a) 3x2 – 2x + 5 remainder 0 3a) 4x2 – 9 remainder 0 5a) 3x2 + 2x + 12 remainder 0 7a) 5x2 – 10x + 26 remainder 44 9a) x2 + 3x + 9 remainder of 0 11a) (-1) is a zero, there are others 13a) 3 is a zero 15a) (-2) and (1) are zeros 17a) (-5) and (1) are zeros 19a) (-1) and (1) are zeros 21) x = -1, 2, ±2𝑖 27) x = -3, -2, 1±𝑖√3 2 1b) 3𝑥 3 − 17𝑥 2 + 15𝑥 − 25 = (x-5)(3x-5)(x+1) 3b) 4𝑥 3 + 8𝑥 2 − 9𝑥 − 18= (x+2)(2x+3)(2x-3) 5b) 3x3 – 16x2 – 72 = (x-6)(3x2 + 2x + 12) 7b) not applicable, remainder is not 0 9b) (x-3)(x2 + 3x + 9) 11b) f(x) = (x+1)(x-2)(x+3) 13b) f(x) = (x-3)(x-3)(2x-1) or (x-3)2(2x-1) 15b) f(x) = (x+2)(x-1)(x-1)(x+1) or (x-1)2(x+2)(x+1) 17b) f(x) = (x+5)(x-1)(2x+3)(x+3) 19b) f(x) = x4 + 7x2 – 8 = (x+1)(x-1)(x2 + 8) 3 23) x = -2, ± 2 25) x = -3, 2, -1, -3/2 29) x = -1, 1, ±2𝑖√2 Section 5.3 1) 𝑓(𝑥) = 𝑥 3 − 8𝑥 2 − 11𝑥 + 148 5) 𝑓(𝑥) = 𝑥 4 − 10𝑥 3 + 30𝑥 2 − 40𝑥 + 104 9) 𝑓(𝑥) = 𝑥 4 + 𝑥 3 − 8𝑥 2 + 4𝑥 − 48 1 3) f(x) = 𝑥 3 − 8𝑥 2 + 22𝑥 − 20 7) 𝑓(𝑥) = 𝑥 4 + 34𝑥 2 + 225 11) x = 2i, -2i, 4 you may write ±2𝑖, 4 13) x = ±2𝑖, 2 , −3 15) x = 5 + 2i, 5 – 2i, -3 17) x = -4+i, -4 – i, 1 19) x = 3+2i, 3-2i, 2 Section 5.4 1a) domain all real numbers except (-3) 1b) vertical asymptote equation x = -3 3a) domain all real numbers except -1 and 4 3b) answer: vertical asymptote x = -1 and x = 4 5a) domain all real numbers except 2, -2 5b) vertical asymptote x = 2 and x = -2 7a) domain all real numbers except 0 and 1 7b) vertical asymptote x = 0 and x = 1 9a) domain all real numbers except 6 9b) answer: vertical asymptote x = 6 11a) domain all real numbers except 3 and -3 11b) vertical asymptote x = 3 and x = -3 13a) domain all real numbers 13b) vertical asymptote none 15a) domain all real numbers 15b) vertical asymptote none 17) x-intercept (3,0) y-intercept (0,-2) 19) x-intercept (-4,0) y-intercept (0,-3) 21) x-intercept (-2/3,0) (4,0) y-intercept (0,2) 23) x-intercept (-4,0) (4,0) y-intercept none Section 5.4 continues 25) x-intercept none y-intercept (0, -1/2) 27) x-intercept (0,0) y-intercept (0,0) 29) x-intercept (-2,0) y-intercept (0, 1/8) 31) x-intercept (0,0) y-intercept (0,0) 33) y = 2 35) y = 0 37) y = 3 41) y = 0 43) y = 0 45) y = 0 51) y = 3x 53) 𝑦 = 2 𝑥 1 49) 𝑦 = 2 𝑥 57) y = x Section 5.5 1) a) domain: all real numbers except 6 b) vertical asymptote x = 6 c) horizontal asymptote y = 3 d) x-intercept (-2,0) e) y-intercept (0,-1) window used x-min = -30 x-max = 30 y-min = -30 y-max = 30 1 39) y = 1 5 47) 𝑦 = 4 1 55) 𝑦 = 3 𝑥 Section 5.5 3) a) domain: all real numbers except 1, -1 b) vertical asymptotes x = 1 x = -1 c) horizontal asymptote y = 4 d) x-intercept (3/2, 0) (-3/2,0) e) y-intercept (9,0) window used x-min -10 x-max 10 y-min -10 y-max 20 5) a) domain:all real numbers except 0 b) vertical asymptote x = 0 c) horizontal asymptote y = 0 d) x-intercept none e) y-intercept none window used x-min -5 x-max 5 ymin -5 y-max 5 Section 5.5 7) a) domain:all real numbers except 4, -4 b) vertical asymptotes x = 4 x = -4 c) horizontal asymptote y = 0 d) x-intercept (-3,0) e) y-intercept (0, =3/16) window x-min -10 x-max 10 y-min -10 ymax 10 9) a) domain: all real numbers except 5 b) vertical asymptote x = 5 c) horizontal asymptote y = 3x d) x-intercept (-5/3,0) (-1,0) e) y-intercept (0,-1) window used x-min -50 x-max 50 ymin -200 y-max 200 Section 5.5 11) a) domain: all real numbers except 1/4 b) vertical asymptote x = 1/4 1 c) horizontal asymptote 𝑦 = 𝑥 2 d) x-intercept (0,0) e) y-intercept (0,0) window used x-min -10 x-max 10 ymin -10 y-max 10 13) a) domain: all real numbers except -3 b) vertical asymptote x = -3 c) horizontal asymptote y = 3/2 d) x-intercept (4,0) e) y-intercept (0,-2) window used x-min -20 x-max 20 y-min 20 y-max 20 Section 5.5 15) a) domain: all real numbers except 0 b) vertical asymptote x = 0 c) horizontal asymptote y = 0 d) x-intercept none e) y-intercept none window: x-min -10 x-max 10 y-min -10 y-max 10 17) a) domain: all real numbers except 0 b) vertical asymptote x = 0 c) horizontal asymptote y = 0 d) x-intercept (-5,0) e) y-intercept none window used x-min -10 x-max 10 ymin -2 y-max 2 this window shows what happens around the x-intercept well, but doesn’t show the vertical asymptote effect very well. I can’t find a window to show both. Section 5.6 1) - 7 < x < 1 3) 1 < x < 5 5) −10 ≤ 𝑥 ≤ 1 7) −1 ≤ 𝑥 ≤ 6 9) x>1 or x < -7 11) x > 5 or x < 1 13) 𝑥 ≥ 1 𝑜𝑟 𝑥 ≤ −10 15) 𝑥 ≥ 1 𝑜𝑟 𝑥 ≤ −6 17) 3 < x < 6 19) -5 < x < 6 21) x > 7 or x < 3 23) x > -1 or x < -5 25) -2 < x < 4 Chapter 5 review 1a) (5,0) even multiplicity, (3,0) odd multiplicity 1b) graph touches x-axis at (5,0) graph crosses x-axis at (3,0) 1c) at most 2 turning points 1d) graph will behave like f(x) = x3 for large values of x. 1e) falls to the left, rises to the right 1f) sketch a graph and approximate the turning points, also label the x-intercepts y 8 6 4 2 -9 -8 -7 -6 -5 -4 -3 -2 (3.67,1.19) 1 (3,0) 2 3 (5,0) 4 5 -1 x 6 7 8 9 -2 -4 -6 -8 1g) increasing (−∞, 𝟑. 𝟔𝟕) ∪ (𝟓, ∞) decreasing (3.67, 5) 2a) (-4,0) odd mult (4,0) odd mult (7,0) odd mult 2b) crosses at (-4,0) crosses at ( 4,0) crosses at ( 7,0) 2c) will have at most 2 turning points. 2d) will behave like f(x) = 3x3 2e) fall left and rises to the right 2f) sketch a graph and approximate the turning points, also label the x-intercepts y 450 400 (-.95,360) 350 300 250 200 150 100 50 -7 (-4,0) -5 -4 -6 -3 -2 -1 1 2 -50 (4,0) 3 4 5 (7,0) 6 7 x 8 9 (5.6,-64.5) 2g) increasing (−∞, −. 𝟗𝟓) ∪ (𝟓. 𝟔, ∞) decreasing (-.95, 5.6) 3a) (0,0) has even multiplicity and both (-3,0) (7,0) have odd multiplicity 3b) graph will cross x-axis at (-3,0) and (7,0) and touch at (0,0) 3c) at most 3 turning points. 3d) will behave like f(x) = x4 3e) rises to the left, rises to the right 3f) sketch a graph and approximate the turning points, also label the x-intercepts y 400 300 200 100 -7 (-3,0) (0,0) -4 -3 -2 -1 (-2.07,-36.14) -6 -5 1 2 3 4 5 (7,0) 6 7 x 8 9 -100 -200 -300 -400 (5.07,-400.36) 3g) 𝒅𝒆𝒄𝒓𝒆𝒂𝒔𝒊𝒏𝒈 (−∞ − 𝟐. 𝟎𝟕) ∪ (𝟎, 𝟓. 𝟎𝟕) 𝒊𝒏𝒄𝒓𝒆𝒂𝒔𝒊𝒏𝒈 (−𝟐. 𝟎𝟕, 𝟎) ∪ (𝟓. 𝟎𝟕, ∞) 4a) I graphed and found the graph crossed the x-axis at x = 3. I will do my synthetic division with x =3, but you could also use -1 or -3 4b) : x3 + x2 – 9x– 9 =(x-3)(x+3)(x+1) 5a) Graph crosses x-axis at x = 1 and x = 2. I will use double synthetic division 5b) x4 +3x3 -8x2 -12x +16 = (x-1)(x-2)(x+2)(x+4) 7) x = -1, ±𝒊√𝟐 6) x = 2,-2,3,-3 8) f(x) = x3 – 4x2 – 2x + 20 9) f(x) = x3 + 5x2 + 16x + 80 10a) all real numbers except 3 10d) (-3,0) 10b) x = 3 10c) y = 2 10e) (0, -2) 10f) y 8 6 4 2 x -18 -16 -14 -12 -10 -8 -6 -4 -2 2 (-3,0) 4 6 8 10 12 14 16 18 -2 (0,-2) -4 -6 -8 11a) all real numbers except -7,3 11b) vertical asymptotes x = -7 and x = 3 11c) y = 0 (the x-axis) 11d) (1,0) 11e) (0, 1/21) 11f) y 4 3 2 1 x -14 -13 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 -1 -2 -3 -4 12a) all real numbers except 5 12b) vertical asymptote x = 5 12c) y = 2x slant asymptote 12d) (-5/2,0) (-1.0) 12e) (-1,0) 12f) 13) - 8 < x < 2 16) x > 2 or x < -4 14) x > 3 or x<-8 15) 4 < x < 6